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Chapter 10: Problem 28

Use a computer algebra system to graph the vector-valued function and identify the common curve. $$ \mathbf{r}(t)=-\sqrt{2} \sin t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{2} \sin t \mathbf{k} $$

Short Answer

Expert verified
The graph of the given vector-valued function forms a helix around the y-axis with the help of a Computer Algebra System.

Step by step solution

01

Identify components

First, separate the vector-valued function \(\mathbf{r}(t)\) into its x, y, and z components. In this case, these components are: \(x(t) = -\sqrt{2} \sin t\), \(y(t) = 2 \cos t\), and \(z(t) = \sqrt{2} \sin t\).
02

Enter function into a computer algebra system

Using a computer algebra system like Mathematica, Matlab, or an online graphing tool suitable for three dimensions, enter each of the components from Step 1 as a function of \(t\). Typically, this involves specifying each coordinate of the function as a separate expression.
03

Graph the vector-valued function

Graph the function using the graphing capabilities of the computer algebra system. Make sure to properly set the limit for the variable \(t\) to get the most accurate graph.
04

Identify the common curve

Once the graph has been generated, observe and identify the common curve in the graph. The shape will depend on the specific function, but it could be a line, plane, spiral, or any number of other shapes. In this case, due to the presence of sine and cosine functions, it is expected to be a helix around the y-axis.

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Most popular questions from this chapter

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} \\ \mathbf{v}(1)=5 \mathbf{j}, \quad \mathbf{r}(1)=\mathbf{0} \end{array} $$

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k} $$

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle $$

A projectile is launched with an initial velocity of 100 feet per second at a height of 5 feet and at an angle of \(30^{\circ}\) with the horizontal. (a) Determine the vector-valued function for the path of the projectile. (b) Use a graphing utility to graph the path and approximate the maximum height and range of the projectile. (c) Find \(\mathbf{v}(t),\|\mathbf{v}(t)\|,\) and \(\mathbf{a}(t)\) (d) Use a graphing utility to complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{t} & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\ \hline \text { Speed } & & & & & & \\ \hline \end{array} $$ (e) Use a graphing utility to graph the scalar functions \(a_{\mathbf{T}}\) and \(a_{\mathrm{N}} .\) How is the speed of the projectile changing when \(a_{\mathrm{T}}\) and \(a_{\mathbf{N}}\) have opposite signs?

In Exercises 39 and \(40,\) find the angle \(\theta\) between \(r(t)\) and \(r^{\prime}(t)\) as a function of \(t .\) Use a graphing utility to graph \(\theta(t) .\) Use the graph to find any extrema of the function. Find any values of \(t\) at which the vectors are orthogonal. $$ \mathbf{r}(t)=3 \sin t \mathbf{i}+4 \cos t \mathbf{j} $$
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